偏微分方程的前向不确定性量化（UQ）是一项需要大量模型评估的多查询任务。本工作的目标是减轻微循环 3D-1D 多尺度计算模型 UQ 的计算成本。为此，我们提出了一种深度学习增强的多保真度蒙特卡洛（DL-MFMC）方法，该方法将多尺度全阶模型（FOM）的信息与深度学习增强的、基于非侵入式投影的降阶模型（ROM）的信息相结合。后者利用本征正交分解（POD）和网格信息神经网络（由作者及合作者先前开发）构建，整合了多种架构来近似 POD 系数，同时引入对微观结构的精细尺度修正。DL-MFMC 方法提供了特定关注量及其相关不确定性的稳健估计器，并实现了计算资源的最优管理。具体而言，计算预算在训练和采样之间高效分配，确保了充分利用 ROM 加速的可靠估计过程。在此，我们应用 DL-MFMC 技术来加速估计关于氧气传输和放射治疗结果的生物物理量。与经典蒙特卡洛方法相比，所提出的方法显示出显著的加速效果，并大幅降低了整体计算成本。
英文摘要：
Forward uncertainty quantification (UQ) for partial differential equations is a many-query task that requires a significant number of model evaluations. The objective of this work is to mitigate the computational cost of UQ for a 3D-1D multiscale computational model of microcirculation. To this purpose, we present a deep learning enhanced multi-fidelity Monte Carlo (DL-MFMC) method that integrates the information of a multiscale full-order model (FOM) with that coming from a deep learning enhanced non-intrusive projection-based reduced order model (ROM). The latter is constructed by leveraging on proper orthogonal decomposition (POD) and mesh-informed neural networks (previously developed by the authors and co-workers), integrating diverse architectures that approximate POD coefficients while introducing fine-scale corrections for the microstructures. The DL-MFMC approach provides a robust estimator of specific quantities of interest and their associated uncertainties, with optimal management of computational resources. In particular, the computational budget is efficiently divided between training and sampling, ensuring a reliable estimation process suitably exploiting the ROM speed-up. Here, we apply the DL-MFMC technique to accelerate the estimation of biophysical quantities regarding oxygen transfer and radiotherapy outcomes. Compared to classical Monte Carlo methods, the proposed approach shows remarkable speed-ups and a substantial reduction of the overall computational cost.
Context-aware learning of hierarchies of low-fidelity models for multi-fidelity uncertainty quantification
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Technical University of Munich（慕尼黑工业大学）
Structure and asymptotic preserving deep neural surrogates for uncertainty quantification in multiscale kinetic equations
University of Ferrara（费拉拉大学）；Italian Ministry of University and Research（意大利大学与研究部）；Royal Society（皇家学会）；European Union（欧盟）；ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing（ICSC – 高性能计算、大数据和量子计算研究中心）；Italian National Institute of High Mathematics（意大利国家高等数学研究所）
A comparative analysis of metamodels for 0D cardiovascular models, and pipeline for sensitivity analysis, parameter estimation, and uncertainty quantification
Inria, Research Center Saclay Ile-de-France（法国国家信息与自动化研究所，萨克雷法兰西岛研究中心）
